Singularities in Boundary Value Problems: Proceedings of the NATO Advanced Study Institute held at Maratea, Italy, September 22 – October 3, 1980

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H.G. Garnir
Springer Science & Business Media, Mar 31, 1981 - Mathematics - 377 pages
The 1980 Maratea NATO Advanced Study Institute (= ASI) followed the lines of the 1976 Liege NATO ASI. Indeed, the interest of boundary problems for linear evolution partial differential equations and systems is more and more acute because of the outstanding position of those problems in the mathematical description of the physical world, namely through sciences such as fluid dynamics, elastodynamics, electro dynamics, electromagnetism, plasma physics and so on. In those problems the question of the propagation of singularities of the solution has boomed these last years. Placed in its definitive mathematical frame in 1970 by L. Hormander, this branch -of the theory recorded a tremendous impetus in the last decade and is now eagerly studied by the most prominent research workers in the field of partial differential equations. It describes the wave phenomena connected with the solution of boundary problems with very general boundaries, by replacing the (generailly impossible) computation of a precise solution by a convenient asymptotic approximation. For instance, it allows the description of progressive waves in a medium with obstacles of various shapes, meeting classical phenomena as reflexion, refraction, transmission, and even more complicated ones, called supersonic waves, head waves, creeping waves, •••••• The !'tudy of singularities uses involved new mathematical concepts (such as distributions, wave front sets, asymptotic developments, pseudo-differential operators, Fourier integral operators, microfunctions, ••• ) but emerges as the most sensible application to physical problems. A complete exposition of the present state of this theory seemed to be still lacking.
 

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Contents

Sur le Comportement Semi Classique du Spectre et de lAmplitude de Diffusion dun Hamiltonien Quantique
1
General InitialBoundary Problems for Second Order Hyperbolic Equations
19
Note on a Singular InitialBoundary Value Problem
55
PseudoDifferential Operators of Principal Type
69
Mixed Problems for the Wave Equation
97
Microlocal Analysis of Boundary Value Problems with Applications to Diffraction
121
Transformation Methods for Boundary Value Problems
133
Propagation of Singularities and the Scattering Matrix
169
Propagation at the Boundary of Analytic Singularities
185
Lower Bounds at Infinity for Solutions of Differential Equations with Constant Coefficients in Unbounded Domains
213
Analytic Singularities of Solutions of Boundary Value Problems
235
Diffraction Effects in the Scattering of Waves
271
Singularities of Elementary Solutions of Hyperbolic Equations with Constant Coefficients
317
The Mixed Problem for Hyperbolic Systems
327
INDEX
371
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